The ever-growing installation of solar power systems imposes severe challenges on the operations of local and regional power grids due to the inherent intermittency and variability of ground-level solar irradiance. In recent decades, solar forecasting methodologies for intra-hour, intra-day and day-ahead energy markets have been extensively explored as cost-effective technologies to mitigate the negative effects on the power grids caused by solar power instability. In this work, the progress in intra-hour solar forecasting methodologies are comprehensively reviewed and concisely summarized. The theories behind the forecasting methodologies and how these theories are applied in various forecasting models are presented. The reviewed mathematical tools include regressive methods, stochastic learning methods, deep learning methods, and genetic algorithm. The reviewed forecasting methodologies include data-driven methods, local-sensing methods, hybrid forecasting methods, and application orientated methods that generate probabilistic forecasts and spatial forecasts. Furthermore, suggestions to accelerate the development of future intra-hour forecasting methods are provided.